PHAS0036
Exam Paper
Exam Mock Exam
[Prof. S. Zane]
Answer ALL THREE questions
Estimated time to complete the paper: 3 hours
This is a MOCK Exam, re-adapted and slightly reduced in content
from the 2020-2021 Exam
(time to completion reduced from 3.5h to 3h)
The numbers in square brackets show the provisional allocation of
maximum marks per question or part of question.
The following data may be used if required:
Speed of light in vacuum, c = 2.998× 108 m · s−1
Permittivity of free space, 0 = 8.85× 10−12 F ·m−1
Permeability of free space, µ0 = 4pi× 10−7 H ·m−1
Gravitational constant: G = 6.674× 10−11 N m2 kg−2
Planck constant: h = 6.626× 10−34 J s
Boltzmann constant: k = 1.381× 10−23 J K−1
Hydrogen mass: mH = 1.674× 10−27 kg
Solar mass: MSun = 1.989× 1030 kg
Solar luminosity: LSun = 3.827× 1026 W
Solar radius: RSun = 6.957× 108 m
Solar effective temperature: T Suneff = 5772 K
PHAS0036/Mock Exam CONTINUED
[Part marks]
1. (a) Photonionization is a process in which a photon is absorbed
by an atom (or molecule) and an electron is ejected. An
energy at least equal to the ionization potential is required.
Let this energy be hν0, and let σν be the cross section for
photoionization. Show that the number of photoionizations
per unit volume per unit time is
4pina
∫ ∞
ν0
σνJν
hν
dν = cna
∫ ∞
ν0
σνuν
hν
dν
where na is the number density of atoms, c is the speed of
light, h is the Planck constant, Jν and uν are the mean
intensity and the energy density of the radiation. [3]
(b) A star of radius R, located at a distance d away, radiates
X–ray photons at a uniform rate Γ (photons per unit volume
per unit time). Neglect photon absorption (i.e. assume an
optically thin medium). A detector at Earth has an angular
acceptance beam of half angle δθ, and an effective area ∆A.
i. Assume that the star is completely resolved. What is the
observed intensity (photons per unit time per unit area
per steradian) toward the centre of the star? [2]
ii. Assume that the star is completely unresolved. What is
the observed average intensity when the star is in the
beam of the detector? [2]
iii. Is the observed intensity larger if the star is resolved or
unresolved? Explain why. [2]
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PHAS0036/Mock Exam CONTINUED
1
[Part marks]
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(c) Consider a plane-parallel slab of a star atmosphere, in which
scattering and absorption and emission occur. Set αν [m−1]
the absorption coefficient, and σ = nσT [m−1] the absorption
coefficient per scattering (also called scattering coefficient).
Here n is the particle number density, and σT is the Thomson
cross section. Idealize the medium as homogeneous and
isothermal, so that σ, αν, n and the temperature do not vary
with depth, and assume scattering is isotropic.
i. Starting from the equation of radiative transfer, derive the
radiative diffusion equation (RDE) in the form
1
3
∂2Jν
∂τ2
= (Jν − Bν)
where Jν is the mean intensity, Bν is the Planck function,
τ is the total optical depth, and ≡ αν/(σ + αν) is the
probability per interaction that a photon will be absorbed.
State all assumptions. [4]
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2
[Part marks]
(Question continued from previous page)
ii. Set τ∗ =
√
3τa (τa + τs), where τa ans τs are the optical
depths for absorption and scattering. Using the RDE with
two–stream boundary conditions (assuming radiation
propagates at two fixed angles, µ = ±1/√3), and using
the Eddington approximation, find the expressions for the
mean intensity Jν(τ) in the slab, for the emergent flux
Fν(0), and for the emergent specific intensity Iν(0) in
terms of Bν, τ∗ and only. [5]
iii. Show that Jν approaches the black body intensity at a
rate governed by an effective optical depth of order τ∗. [2]
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3
[Part marks]
2. (a) A star in hydrostatic equilibrium, has a ratio of specific heats
γ = 5/3, and satisfies the equation of state for an ideal gas. It
is just stable against convection. Show that the radiative
temperature gradient at radius r in the star is given by∣∣∣∣dTdr
∣∣∣∣
rad
=
2µmH Gm(r )
5kr 2
where m(r ) is the mass contained within radius r , µ is the
mean molecular weight of material, k is Boltzmann’s
constant, mH is the hydrogen atomic mass, and G is the
universal gravitational constant. [2]
(b) Explain the role of the scale height in hydrostatic equilibrium.
Calculate the atmospheric scale heights for the Sun, a
solar-mass red giant (R = 100 RSun, Teff = 3000K), and a 10
MSun red supergiant (R = 1000 RSun,Teff = 3000K). Give your
answers in units of km, and as a fraction of the respective
stellar radii. [3]
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4
[Part marks]
(Question continued from previous page)
(c) In terms of the Lane–Emden variables θ and ξ, the mass of a
polytropic star can be written as
M = 4piα3
{
K (n + 1)
4piGα2
}n/(n−1){
−ξ2dθ
dξ
∣∣∣∣
ξ1
}
where α = R/ξ1, R is the star radius, n is the polytropic index,
and K is a constant.
i. Briefly introduce the hypothesis at the basis of the
“Standard Eddington Model” (SEM). Set P and ρ the total
pressure and the density. Using the equation of state for
a SEM, P = Kρ4/3, demonstrate that in this case the
above relation becomes
M ≈ 18MSun
µ2
(1− β)1/2
β2
where β = PG/P = (1− L/LEdd ), µ is the mean molecular
weight, PG is the gas pressure, L is the star’s luminosity,
and LEdd is the Eddington limit. (Note: you do not need to
derive the numerical coefficient, just the dependence of
the mass on µ and β). Draw an inference on the relative
roles of gas pressure and radiation pressure with
increasing mass, explaining your reasoning. [5]
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5
[Part marks]
(Question continued from previous page)
ii. Demonstrate that, in the SEM, the mass-luminosity
relation is
L ∝ β4µ4M3 .
Demonstrate that the above relation can be approximated
as L ∝ M3 for stars with M ∼ MSun, while it flattens to
L ∝ M for massive stars (M ∼ 100MSun). [6]
iii. Although the SEM does not in itself know anything about
nuclear physics and thus about stellar evolution, it
nonetheless makes useful predictions for stellar
evolution. A star with a composition similar to the
present-day Sun has µ = 0.61, which is for a composition
where the medium contains three times as much
hydrogen, by mass, compared to helium. Suppose that
the star evolves and at some point has burned much of
its hydrogen into helium, so the ratio is reversed, and
µ = 0.91. What does this do to its luminosity? Give your
answer for two cases, assuming the star mass is
M = MSun or M = 100MSun. Assume the star mass is
constant during the evolution, and use the results you
have derived in the previous points. [4]
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6
[Part marks]
3. (a) Explain the concept of ‘homology’, and describe the
circumstances under which it may reasonably be applied to
stars. [2]
(b) The densities and pressures of two homologous stars of
masses M1 and M2 are related by
ρ2
ρ1
=
(
M2
M1
)(
R1
R2
)3
and
P2
P1
=
(
M2
M1
)2(R1
R2
)4
.
By using these equations, and starting with the equation of
state for an ideal gas, show that
T1
T2
=
µ1
µ2
M1
M2
R2
R1
.
(where µ is the mean molecular weight). [4]
(c) Supposing that the energy-generation rate per unit mass is
given by
ε(r ) = ε0ρ(r )T n(r )
(where ε0 is a constant), use the equation of continuity of
energy to show that
L2
L1
=
(
ε0,2
ε0,1
)(
µ2
µ1
)n(M2
M1
)(n+2)(R1
R2
)(n+3)
[6]
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7
[Part marks]
(Question continued from previous page)
(d) The equation of radiative energy transport is
dT
dr
= − 3
16pi
κR(r )ρ(r )
r 2
L(r )
acT 3
.
where the opacity may be assumed to be given by
κR(r ) = κ0ρ(r )T−m(r ).
Show that(
ε0,1
ε0,2
)(
κ0,1
κ0,2
)(
µ1
µ2
)(n−m−4)(M1
M2
)(n−m)(R2
R1
)(n−m+6)
= 1.
[4]
(e) Hence obtain mass–radius and mass–luminosity
relationships for low-mass stars, for which the power-law
exponents may be taken to be m = 3.5,n = 5. [4]
PHAS0036/Mock Exam END OF EXAMINATION PAPER

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